Triply periodic constant mean curvature surfaces

TL;DR

This paper establishes area bounds for closed surfaces with constant mean curvature in a flat 3-torus, highlighting differences from minimal surface cases and contributing to geometric analysis.

Contribution

It provides new area estimates for constant mean curvature surfaces of any genus in flat 3-tori, contrasting with known minimal surface results.

Findings

Area estimates depend on mean curvature and genus

Contrasts with Traizet's minimal surface theorem

Shows existence of bounded-area CMC surfaces in flat tori

Abstract

Given a closed flat 3-torus $N$, for each $H>0$ and each non-negative integer $g$, we obtain area estimates for closed surfaces with genus $g$ and constant mean curvature $H$ embedded in $N$. This result contrasts with the theorem of Traizet [33], who proved that every flat 3-torus admits for every positive integer $g$ with $g\neq 2$, connected closed embedded minimal surfaces of genus $g$ with arbitrarily large area.